Navigating K-Nearest Neighbor Graphs to Solve Nearest Neighbor Searches

Nearest neighbor queries can be satisfied, in principle, with a greedy algorithm under a proximity graph. Each object in the database is represented by a node, and proximal nodes in this graph will share an edge. To find the nearest neighbor the idea is quite simple, we start in a random node and get iteratively closer to the nearest neighbor following only adjacent edges in the proximity graph. Every reachable node from current vertex is reviewed, and only the closer-to-the-query node is expanded in the next round. The algorithm stops when none of the neighbors of the current node is closer to the query. The number of revised objects will be proportional to the diameter of the graph times the average degree of the nodes. Unfortunately the degree of a proximity graph is unbounded for a general metric space [1], and hence the number of inspected objects can be linear on the size of the database, which is the same as no indexing at all.

In this paper we introduce a

quasi

-proximity graph induced by the all-

k

-nearest neighbor graph. The degree of the above graph is bounded but we will face local minima when running the above greedy algorithm, which boils down to have false positives in the queries.

We show experimental results for high dimensional spaces. We report a recall greater than 90% for most configurations, which is very good for many proximity searching applications, reviewing just a tiny portion of the database.

The space requirement for the index is linear on the database size, and the construction time is quadratic in worst case. Relaxations of our method are sketched to obtain practical subquadratic implementations.